鎖状多体系の末端における活発な運動と遅い緩和 (非線形波動現象の多様性と普遍性)

Size: px

Start display at page:

Download "鎖状多体系の末端における活発な運動と遅い緩和 (非線形波動現象の多様性と普遍性)"

さやなまきい
5 years ago
Views:

1 (Tetsuro KONISHI) (Tatsuo YANAGITA2) 1 Department of Physics, Nagoya University 2 Research Institute for Electronic Science, Hokkaido University ( ) ( ) $k$ $k$ ( ) ( ) Boltzmann-Jeans $k$ $\exp(c\sqrt{k})$ : ( )

2 $\bullet$ $\bullet$ $\bullet$ //www ( ) $?\cdots\cdots\cdots\cdots\cdots\cdot\cdot$ $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ Boltzmann-Jeans DNA 2 : $*1_{o}$ (3 ):http: $//www$.youtube. $com/watch?v=alr_{-}aq$-jhry (YouTube) $:$ (7 ) : http (YouTube) youtube. om $/wat$ ch?v XSuQYRRkRyQ $c$ $=$ $N$ $(N=1,2, \ldots)$ 1 $N\geq 2$ [OY98, SOY99, SY00]: $*1$ 3 2

3 $\bullet$ $N$ $m_{i}$, $i=1,2,$ $\cdots,$ $N$ $(i$ $\ell_{i})$ ( g) $i$ $(x_{i}, y_{i})$ (0,0) 1 $(\begin{array}{l}x_{i}y_{i}\end{array})=(\begin{array}{l}x_{i-1}y_{i-l}\end{array})+\ell_{i}(\begin{array}{l}sin\varphi_{i}-cos\varphi_{i}\end{array})=\sum_{j=1}^{i}\ell_{j}(\begin{array}{l}sin\varphi_{j}-cos\varphi_{j}\end{array})$ $\infty upkd$ pendulum: slze-7 time-averaged $kinet\dot{c}$ energy of each pendulum $0$ $5\infty$ $\epsilon\infty$ O00 $r$ e 2 7 $K_{i}(t)\equiv mv_{i}(t)^{2}/2$ $\overline{k_{i}(t)}\equiv(1/t)\int_{0}^{t}k_{i}(t )dt $ 3

4 103 : $L= \sum_{i=1}^{n}\frac{m_{i}}{2}(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})-\sum_{i=1}^{n}m_{i}gy_{i}$ (1) $= \sum_{j=1}^{n}\sum_{k=1}^{n}(\frac{1}{2}\sum_{i=\max(j,k)}^{n}m_{i})l_{j}l_{k}\dot{\varphi}_{j}\dot{\varphi}_{k}\cos(\varphi_{j}-\varphi_{k})$ $+ \sum_{i=1}^{n}m_{i}g\sum_{j=1}^{i}l_{j}\cos\varphi_{j}$ (2) $t$ $K_{i}(t) \equiv\frac{mv_{i}(t)^{2}}{2}$ $\overline{k_{i}(t)}\equiv$ $\frac{1}{t}\int_{0}^{t}k_{i}(t )dt $ 3 $mv^{2}/2$ [OY98, $SOY99$, SYOO]. $T$ 1 $= \frac{1}{2}k_{b}t$ (3) $H(q,p)= \sum_{i=1}^{n}\frac{m_{i}}{2}v_{i}^{2}+u(\{q_{i}\})$ (4) $\{\frac{m_{i}v_{i}^{2}}{2}\rangle$ $\sim\vee$ o $H(q,p)=K(q,p)+V(q),$ $K(q,p)= \frac{1}{2}\sum_{i,j}\alpha_{ij}(q)p_{i}p_{j}$ (5) 4

5 104 $\{K_{i}^{(c)}\}\equiv\langle\frac{1}{2}p_{i}\frac{\partial K}{\partial p_{i}}\rangle=\frac{1}{2}k_{b}t$ (6) $i$ [To118, To138, 61] $m\dot{q}_{i}$ $q_{i}$ (2) $K= \frac{1}{2}\sum_{i,j})\dot{\varphi}\iota \mathscr{s}\equiv\frac{1}{2}\vec{\mathscr{s}}{}^{t}a(\varphi)\vec{\mathscr{s}}$ (7) $\varphi$ $p_{n} \equiv\frac{\partial L}{\partial \mathscr{s}_{n}}=\ell_{n}\sum_{k=1}^{n}(\sum_{i=\max(k,n)}^{n}m.)\ell_{k\dot{\varphi}_{k}\cos(\varphi_{n}-\varphi_{k})\equiv\sum_{k=1}^{n}a(\varphi)_{nk}\mathscr{s}k}$ $\vec{p}=a(\varphi)\vec{\dot{\varphi}}$ $K= \sum_{i,j=1}^{n}\frac{1}{2}a(\varphi)_{ij}\dot{\varphi}_{i}\dot{\varphi}_{j}\equiv\frac{1}{2}\vec{\dot{\varphi}}{}^{t}a(\varphi)\vec{\dot{\varphi}}=\sum_{\dot{\iota},j=1}^{n}\frac{1}{2}(a^{-1}(\varphi))_{ij}p_{i}p_{j}\equiv\frac{1}{2}\vec{p}{}^{t}a^{-1}(\varphi)\vec{p}$. (5) 3 $\overline{\frac{1}{2}mv^{2}}$ 4 $mv^{2}/2$ 5

6 pendulum. linear and-canonical kinetic energy site 3 $\overline{k_{i}}(+$ $(N=12)$ $)$ $\overline{k_{i}^{(c)}}(\cross$ $\overline{1/2mv^{2}}$ $)$ 4Planar chain ) 5 ( ) 5.1 Plnanar chain model [KY09, Kra46, Maz96] $N$ $i=1,2,$ $N$ $\cdots,$ $(i$ 3 Kramers [Kra46], freely jointed chain [Maz96] $m_{i}$, 6

7 $i$ 106 $(x_{i}, y_{i})$ : $L= \sum_{i=1}^{n}\frac{m_{i}}{2}(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})-u(\{\vec{r_{i}}\})$, $ \vec{r_{i+1}}-\vec{r_{i}} ^{2}-\ell_{i}^{2}=0$ $(\vec{r}_{i}\equiv(x_{i}, y_{i}))$, $i=1,2\cdots,$ $N-1$. $i$ $(x_{i}, y_{i})$ $(\begin{array}{l}x_{i+1}y_{i+1}\end{array})-(\begin{array}{l}x_{i}y_{i}\end{array})=\ell_{\dot{\iota}}(\begin{array}{l}sin\varphi_{i}-cos\varphi_{i}\end{array})$ (8) $K \equiv\sum_{i=1}^{n}arrow m_{2}\cdot(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})$ $K= \frac{m}{2}(\dot{x}_{g^{2}}+\dot{y}_{g^{2}})+\frac{m}{2}\sum_{j,k=1}^{n-1}\mu_{j}^{\leq}\mu_{k}^{>}\cos(\varphi_{j}-\varphi_{k})\ell_{j^{\ell_{k}\mathscr{s}}j\dot{\varphi}_{k}}$, $M \equiv\sum_{i=1}^{n}m_{i},$ $\mu_{n}^{\leq}\equiv\sum_{k=1}^{n}\mu_{k},$ $\mu_{n}^{>}\equiv\sum_{k=n+1}^{n}\mu_{k},$ $\mu_{k}\equiv\frac{m_{k}}{m}$. $\varphi$ 5.2 $H=E$ 1 $N-1$ [ ] $+$ ] ( ) $\triangleright \mathscr{z}$ 5 ) planar chain model SHAKE RATTLE 2 simplectic integrator 3 4 simplectic integrator tc $[LR04]_{0}$ $H=E$ $L=\ell_{0}$ 7

8 107 planar chain with 16 masses convergence of linear kinetic $energ\ovalbox{\tt\small REJECT} es$ $00e+0$ $20e+4$ $40e+4$ $60e+4$ $80e+4$ $10e+5$ $t_{-}mu$ 6 ( ) $K_{i} \equiv\frac{1}{2}m_{i}(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})$ ( ) [KY09] $N=16$ $\overline{k_{i}},$ 5 ( ) $m_{i}=1$ $(i=1,2, \cdots, N)$. $\ell_{i}=1(i=1,2, \cdots, N)$ $dt=10^{-3}$ $T_{t}$ $tal=10^{5}$ 1 6 $\overline{k_{i}}$ [KYO9] ( 7) 6 7 $\overline{k_{i}^{(c)}}$ 6 8 $\mathfrak{n}$ $16\cdot pium$-chai cmmal kinetic energy $0$ $12$ $i$ [KY09]

9 $T$ $\langle\cdots\}\equiv\frac{1}{z}\int$... $e^{-\beta H}d\Gamma$ (9) $(\beta\equiv 1/k_{B}T)$ $i$ $\langle K_{i}\rangle\equiv\{\frac{m_{i}}{2}(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})\}$ (10) $= \{\frac{m_{i}}{2}(\dot{x}_{g^{2}}+\dot{y}_{g^{2}})\}+\{\frac{m_{i}}{2}(\sum_{j,k=1}^{n-1}a_{ij}a_{ik}\cos(\varphi_{jk})\dot{\varphi}_{j}\mathscr{s}k)\rangle$ $= \frac{m_{i}}{m}k_{b}t+\frac{m_{i}}{2}\sum_{j,k=1}^{n-1}a_{ij}a_{ik}\langle\cos(\varphi_{jk})\mathscr{s}j\dot{\varphi}_{k}\rangle$ (11) $X_{G},$ $Y_{G}$ $X$ $Y$ 2 ( ) $m_{i}=m$ $\langle K_{i}\rangle=\frac{m_{i}}{M}k_{B}T\{1+\frac{1}{2}[\sum_{j=1}^{i-1}(\frac{j}{N-j})+\sum_{j=i}^{N-1}(\frac{N-j}{j})]\}$ (12) $\{K_{1}\rangle>\{K_{2}\rangle>\cdots<\cdots<\langle K_{N-1}\}<\{K_{N}\rangle$ (13) $[KY09]$ $arrow$ $\ovalbox{\tt\small REJECT}$ $\vec{r_{i}}=\vec{r_{g}}+\sum_{j=1}^{n-1}a_{ij}\vec{s_{j}}$ ( 8) $a_{ij}$ 8 $\varphi$ $a$ $i$ $\{K_{i}\rangle$ $a_{ij}$ $mj$ $\langle K_{i}\rangle=\frac{m_{i}}{M}\cdot\frac{D}{2}k_{B}T+k_{B}T\sum_{j=1}^{N-1}\frac{m_{i}a_{ij}^{2}}{\sum_{n=1}^{N}m_{n}a_{nj}^{2}}$ (14) 9

10 109 1 $D$ 1 (12) 6 ( ) 6.1 $\overline{k_{i}}$ $ \vec{r_{i+1}}-\vec{r_{i}} =\ell_{i}$ (15) $K_{i}$ (15) (15) $\{\frac{m_{i}}{2}v_{i}^{2}\rangle=\frac{1}{2}k_{b}t$ (16) $N$ 1 [KY10] 9 spring-chain model. $L= \sum_{i=1}^{n}\frac{m}{2}(\dot{x}_{i}^{2}+\dot{y}_{i}^{2})-\sum_{i=1}^{n-1}\frac{k}{2}\{ \vec{r_{i+1}}-\vec{r_{i}} -l_{i}\}^{2}-\sum_{i=1}^{n}u(\{r_{i}\})$ $\vec{r_{i}}\equiv(x_{i}, y_{i}),$ $U(\vec{r})$ :( ) 10

11 : $N=6$ ( ), 4 symplectic integrator $m_{i}=1$ $(i=1,2, \cdots, N)$, $\ell_{i}=1$ $(i=1,2, \cdots, N)$ $N=8,$ $k=10^{4}$ $\Delta^{(1)}(t)$ $\Delta^{(2)}(t)$ $\langle\cdots\rangle_{n}\equiv\frac{1}{n}\sum_{j=1}^{n}\cdots$ $\Delta^{(1,2)}=0$ [TKG94, TGK96, TGK97, SSTOO, $TMK89$] const. $\Rightarrow$ $\overline{k_{i}}=$ $i$ 11 spring-chain model $N=8,$ $k=10^{4}$ : $\overline{k_{2}(t)}\equiv(1/t)\int_{0}^{t}k_{i}(t )dt $ : 11

12 111 $0$ time $\Delta^{(1)}$ $\Delta^{(1)}(t)\equiv\frac{1}{N}\sum_{i=1}^{N}\ulcorner K_{i}(t)-\langle\overline{K_{j}}(t)\rangle_{N}]^{2}$, $\Delta^{(2)}(t)\equiv\frac{1}{\tau}l^{t+\tau}\frac{1}{N}\sum_{i=1}^{N}[K_{i}(t )-\langle K_{j}(t )\}_{N}]^{\text{ }}dt $ $\overline{k_{i}}(t)\equiv\frac{1}{\tau}l^{t+\tau_{k_{i}}}(t^{l})dt $ $K_{i}(t) \equiv\frac{m}{2}(\dot{x}_{i}^{2}(t)+\dot{y}_{i}^{2}(t))$ $\bullet$ $\triangle(t)$ ( ) $K_{vib}/K_{r}$ $t$ ( ) 12

13 $k$ 14 $k$ $H=K_{vib}(\{i_{i}\})+K_{rot}(\{\dot{\varphi}_{i}\})+H_{int}(\{i_{i}\},$ $\{\mathscr{s}i\})+u$ $K_{vib}$ $K_{rot}$ ( 13) $k$ 14 [KY10]

14 $\tilde{\dot{\phi g}\mathfrak{d}0}$ $\underline{\#\epsilon\xi\underline{v\leftrightarrow \text{ }i}}$ 113 $3-\dim$ spring-chain: size 16: spnng-chain with branch : $N=16,$ $k=10^{4}$ ( ) ( ) ( 16) $i=n$ Boltzmann-Jeans $\tau$ 14 $k$ $*$ Boltzmann-Jeans Boltzmann-Jeans 2 $[Bo195,$ $Jea03,$ $Jea05$, BGG87, BGG89, $SS99$, SIS06, NKOO, $MK05$] $*2$ Boltzmann-Jeans conjecture conjecture 14

15 $a\epsilon Qvv\grave{8}\dot{\iota}>$ $\underline{\underline{v\not\in a}}$ 114 $3-\dim$ spring-chain: size 8: $1\infty p\cdot with$-tail site 16 ) : $N=8,$ $k=10^{4}$ ( ) ( ) $\vdash$ [BGG89] : $\propto\exp(\frac{\tau(\text{ ^{ }})}{\tau(\text{ })})$ (17) $\tau$ ( ) 1/V $\sim$ $\tau$ $\sim$ ( ) 1 ( ) $=$ ( ) (18) $\propto\exp(c\sqrt{k})$ $t_{relax}$ Boltzmann-Jeans $k$ [KYIO] 7 ( ) $k$ $k$ ( ) $k$ Boltzmann-Jeans $\exp(c\sqrt{k})$ 15

16 $\sqrt{k}$ 17 $N=6$, 15 Boltzmann -Jeans : $t_{relax}\sim 5.52\cross 10^{4}\exp(0.415\sqrt{k})[KY10]$ 3 [BGG87] Giancarlo Benettin, Luigi Galgani, and Antonio Giorgilli. Exponential law for the equipartition times among translational and vibratinal degrees of freedom. Physics Letters $A$, Vol. 120, pp ,

17 116 [BGG89] G. Benettin, L. Galgani, and A. Giorgilli. Realization of homoclinic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part II. Comm. Math. Phys., Vol. 121, pp , [Bo195] L. Boltzmann. On certain questions of the theory of gases. Nature, Vol. 51, p. 413, [KY10] [LR04] [Maz96] [MK05] [NKOO] [Jea03] J. H. Jeans. On the vibrations set up in molecules by collisions. Phil. Mag., Vol. 6, p. 279, [Jea05] J. H. Jeans. On the partition of energy between matter and aether. Phil. Mag., Vol. 10, p. 91, [Kra46] H. A. Kramers. The behavior of macromolecules in inhomogeneous flow.. $J$ Chem. Phys., Vol. 14, pp , [KY09] T. Konishi and T. Yanagita. Energetic motions of end-particles in constrained dynamical systems. Journ of Statistical Mechanics, p. L09001, $al$ T. Konishi and T. Yanagita. Slow relaxation to equipartition in spring-chain systems. Journal of Statistical Mechanics, p , B. Leimkurler and S. Reich. Simulating Hamiltonian Dynamics. Cambridge Univ. Press, Martinal Mazars. Statistical physics of the freely joined chain. Phys. Rev. $E$, Vol. 53, pp , Hidetoshi Morita and Kunihiko Kaneko. Roundabout relaxation: Collective excitation requires a detour to equilibrium. Phys. Rev. Lett., Vol. 94, $p$ , Naoko Nakagawa and Kunihiko Kaneko. Energy storage in a hamiltonian system in partial contact with a heat bath. J. Phys. Soc. Jpn., Vol. 69, pp , [OY98] Yoshihito Oyama and Tatsuo Yanagita, talk at the meeting of the [SIS06] [SOY99] Physical Society of Japan. A. Shudo, K. Ichiki, and S. Saito. Origin of slow relaxation in liquid-water dynamics: A possible scenario for the presence of bottleneck in phase space. Europhys. Lett., Vol. 73, pp , N. Saitoh, Yoshihito Oyama, and Tatsuo Yanagita, talk at the meet- 17

18 117 [SS99] [SSTOO] [SYOO] ing of the Physical Society of Japan. Vol. 73, pp , Diane E. Sagnella, John E. Straub, and D. Thirumalai. Time scales and pathways for kinetic energy relaxation in solvated proteins: Application to carbonmonoxy myoglobin. Journal of Chemical Physics, Vol. 113,, N. Saitoh and Tatsuo Yanagita, talk at the meeting of the Physical Society of Japan. [TGK96] T. Tsuchiya, N. Gouda, and T. Konishi. Relaxation processes in onedimensional self-gravitting many-body systems. Phys. Rev, Vol. $E53$, pp , [TGK97] [TKG94] T. Tsuchiya, N. Gouda, and T. Konishi. Chaotic itinerancy and thermalization in one-dimensional self-gravitating systems. Astrophysics and Space Science, Vol. 257, pp , T. Tsuchiya, T. Konishi, and N. Gouda. Quasiequilibria in one-dimensional self-gravitating many body systems. Physical Review, Vol. $E50$, pp , [TMK89] D. Thirumalai, Raymond D. Mountain, and T. R. Kirkpatrick. [To118] Ergodic behavior in supercooled liquids and in glasses. Physical Review $A$, Vol. 39, pp , Richard C. Tolman. A general theory of energy partition with applications to quantum theory. Physical Review, Vol. 11, pp , [ [To138] Richard C. Tolman. The Principles of Statistical Mechanics. Oxford University Press, Oxford, ] 6 [1]

1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)

START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) 2 SCA 2.0 CA ( ) E.g.